A classification of smooth embeddings of 4-manifolds in 7-space, I

We work in the smooth category. Let N be a closed connected n-manifold and assume that m>n+2. Denote by Em(N) the set of embeddings N→Rm up to isotopy. The group Em(Sn) acts on Em(N) by embedded connected summation of a manifold and a sphere. If Em(Sn) is non-zero (which often happens for 2m<3n+4) then until recently no complete readily calculable description of Em(N) or of this action were known (as far as I know). Our main results are examples of the triviality and the effectiveness of this action, and a complete readily calculable isotopy classification of embeddings into R7 for certain 4-manifolds N. The proofs use new approach based on the Kreck modified surgery theory and the construction of a new invariant. Corollary – (a) There is a unique embedding f:CP2→R7 up to isoposition (i.e. for each two embeddings f,f′:CP2→R7 there is a diffeomorphism h:R7→R7 such that f′=h○f).(b) For each embedding f:CP2→R7 and each non-trivial embedding g:S4→R7 the embedding f#g is isotopic to f.